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Positron Emission Tomography » Resolution

 

Resolution and Covariance of PET Images

Jeff Fessler of University of Michigan used some clever approximation methods to develop analytical expressions for the resolution and noise properties of MAP (or equivalently penalized-ML) reconstructions. Subsequent work, both by our group and by Dr. Fessler and colleagues, has extended these results to the point where we can compute, with reasonable accuracy and computational cost, the local impulse response (from which we can infer resolution) and the covariance of MAP reconstructions. Furthermore, as originally suggested by Fessler and Rogers, we can use the same analysis to modify the prior so as to achieve near uniform resolution throughout the image, even in a realistic shift-variant 3D tomogram.

We have derived approximate analytical expressions for the local impulse response and covariance of images reconstructed from fully-3D PET data using MAP estimation. These expressions explicitly account for the spatially variant detector response and sensitivity of a 3D tomograph. The resulting spatially variant impulse response and covariance are computed using 3D Fourier transforms.

 

Figure: Using the local impulse response we can compute the local contrast recovery coefficient (CRCs) at each voxel. The left figure above shows the accuracy of analytically approximated CRCs in comparison to the measured CRCs simulated for the microPET - the plane index are for each of the 8 crystal rings. Notice also that the CRCs drop off towards the edge of the scanner. The right figure shows the effect of applying a spatially variant smoothing prior - the CRCs become approximately shift invariant, and the analytic approximations remain reasonably acccurate.

 

Figure: Variances computed from the theoretical approximations. Note that the predicted values closely follow the Monte Carlo values even at low count regions. To improve accuracy in low count regions in which the nonnegativity constraint may be active, we have developed a method for using a truncated Gaussian distribution to account for the effect on the variance of the non-negativity constraint. The figure at left are variances for the uniform penalty (non-uniform CRC) and the figure at right are for spatially variant penalty (uniform CRCs).

 

Other potential applications of the theoretical analysis of impulse response and covariance include:

  • selection of smoothing parameters to maximize contrast recovery for potential improvements in lesion detectability
  • use of estimated covariances to compute the variance of activity integrated over a region of interest
  • use of the local impulse response and covariance to compute, in closed form, the area under the ROC curve for computer observers designed to reflect human observer performance.